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Omni-MATH / Let P be a polynomial with integer coefficients such that P(0)=0 and

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let

P

be a polynomial with integer coefficients such that

P(0)=0

and

\gcd(P(0), P(1), P(2), \ldots ) = 1.

Show there are infinitely many

n

such that

\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.

Plain-text mathematical notation (without MathML)

Let P be a polynomial with integer coefficients such that P(0)=0 and
gcd(P(0),P(1),P(2),…)=1.
Show there are infinitely many n such that
gcd(P(n)−P(0),P(n+1)−P(1),P(n+2)−P(2),…)=n.

Original LaTeX notation

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and
\[\gcd(P(0), P(1), P(2), \ldots ) = 1.\]
Show there are infinitely many $n$ such that
\[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

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Source and history

task_id: 3e2c8498-8e66-5ea3-ac80-150b0ac26849
task_key: test--3e2c8498-8e66-5ea3-ac80-150b0ac26849
task_revision_id: 2
config / split: / test

Official source

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